Mine were stuck up for a while. Turns out it was just a button on the dash on the right hand side of the steering wheel. Not sure if you pushed that? If you tried that then my best advice is to take it to an auto electrician most will take a look at it on the spot for you. They looked at mine for free and sorted it out pretty quick.

But then comparing to videos I took when dismantling it, I see that I have that extra cord going out of mine.

However, I do remember when I was undoing stuff around the steering wheel that there was another plug that wasn't connected to anything.. maybe I'll take it apart again and try put the headlight stalk foam covered plug into that other one and see what happens

The headlight popup switch might not be working properly, or might not be plugged in correctly. In that case, there are 2 contacts on the plug that need to be connected in order for the lights to operate normally. I don't remember which, but a search should tell you. I had to bridge them when i repurposed that switch. Lights would go up and stay up until I put a paperclip across them, same as your problem.

Tldr; your issue entirely relates to the button on the right of the dash. Plug it in or stick some wire between the pins that normally stay closed. It probably just hasn't been plugged in though.

Thanks for that, it could just be the button is broken. Winding them down manualy and pressing it pops them up but not vice versa.
Which contacts would I have to cross? I didn't really understand the wiring diagram

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You want the two normally-closed contacts bridged, meaning pins 1 and 2 which are normally connected when "OFF".

I'm pretty sure I just ran a continuity test with a multimeter on mine, then bridged whatever two pins were connected when the switch was off. You could do the same and test the switch on/off, or take a guess and bridge what you think is right using a paper clip (namely the top row, far left, 1st and 2nd on the same row, looking at that picture with the locking tab at the top), or open up that switch and clean all the contacts. Chances are the contact connecting to the 2nd pin just needs a scuff up with some sandpaper.